Electrons three-bodied systems

Helium is not the only three body system under study, but it is likely the most complicated one. The electron-electron interaction is comparable with the electron-nucleus interaction and cannot be considered as a perturbation. The situation is different with helium-like (and lithium-like) ions, where the electron-electron interaction is as small as 1 jZ with respect to the interaction of an electron and the nucleus. 

The p and He2+ are thus regarded as two atomic centers in a diatomic molecule. Because of the dual character as an exotic atom and an exotic molecule Antiprotonic Helium is often called antiprotonic helium atom-molecule, or for short, atomcule. Since the Is electron motion, coupled to a large-(n, l) p orbital, is faster by a factor of 40 than the p motion, the three-body system pHe+ is solved by using the Born-Oppenheimer approximation, as fully discussed by Shimamura [6]. 

In this section we derive a set of regularized equations of motion and a triple collision manifold (TCM) for the Coulomb three-body system. Three particles (electron, nucleus, and electron) have masses mi = mg, m2 = m and m3 = mg and charges —e, Ze, and —e. We consider the Coulomb three-body system whose Hamiltonian is... 

Consider a diatomic, AB, interacting with a surface, S. The basic idea is to utilize valence bond theory for the atom-surface interactions, AB and BS> along with AB to construct AB,S For each atom of the diatomic, we associate a single electron. Since association of one electron with each body in a three-body system allows only one bond, and since the solid can bind both atoms simultaneously, two valence electrons are associated with the solid. Physically, this reflects the ability of the infinite solid to donate and receive many electrons. The use of two electrons for the solid body and two for the diatomic leads to a four-body LEPS potential (Eyring et al. 1944) that is convenient mathematically, but contains nonphysical bonds between the two electrons in the solid. These are eliminated, based upon the rule that each electron can only interact with an electron on a different body, yielding the modified four-body LEPS form. One may also view this as an empirical parametrized form with a few parameters that have well-controlled effects on the global PES. 

Within the hyperspherical method, new quantum numbers K, T and A are introduced to describe two-electron correlations. Both K and T are angular correlation numbers (omitted here for simplicity, see [333]), while A = 0, 1 is a radial quantum number, often written as 0,+,— because it is related to the + and — classification of Cooper, Fano and Pratts [323] described in section 7.10. Another quantum number which is often used is v = n — 1 — K — T, where n is the principal quantum number. The number v turns out to be the vibrational quantum number of the three-body system, or the number of nodes contained between the position vectors ri and r2 of the two electrons [334]. 

There have now been several applications reported for fiilly exponentially correlated four-body wavefunctions [7-9], also limited to bases without pre-exponential r,j. While it was found that pre-exponential are relatively unimportant for three-body systems, they can be expected to contribute in a major way to the efficiency of expansions for three-electron systems such as the Li atom and its isoelectronic ions, as is obvious from the fact that the zero-order description of the ground states of such systems has electron configuration s 2s. 

Having two nuclei and one electron, H is a three-body system like the helium atom. As was the case for helium, the Schrodinger equation for Hj cannot be exactly solved by analytical integration, but again we can find tractable and accurate approximations. The coordinates we will use for diatomics are drawn in Fig. 5.1. The two nuclei are labeled A and B and lie on the z axis. The position of the electron is given in spherical coordinates, with the origin at either nucleus A or nucleus B. 

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